Models.heat.Buoyancy Water

8/10/2019 Models.heat.Buoyancy Water

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Solved with COMSOL Multiphysics 4.4

1 | B U O Y A N C Y F L O W I N W A T E R

Buoyancy Flow in WaterIntroduction

This example studies the stationary state of free convection in a cavity filled with waterand bounded by two vertical plates. To generate the buoyancy flow, the plates areheated at different temperatures, bringing the regime close to the transition betweenlaminar and turbulent.

An important step in setting up a convection model is to assess whether the flow stayslaminar or becomes turbulent. It is also important to approximate how fine should bethe mesh needed to resolve velocity and temperature gradients. Both of theseapproximations rely on the velocity scale of the model. This makes the set-up of naturalconvection problems challenging since the resulting velocity is part of the nonlinearsolution. There are, however, tools to approximate scales for natural convectionproblems. These tools are demonstrated in this model using simple 2D and 3D

geometries.

A first 2D model representing a square cavity (see Figure 1 ) focuses on the convectiveflow.

Hot temperature (20 C)

Pressure point

Water

Thermal insulation

Thermal insulationSquare side length (10 cm)

on the right plate

constraint (0 Pa)

Cold temperature (10 C)on the left plate

Figure 1: Domain geometry and boundary conditions for the 2D model (square cavity).


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The 3D model (see Figure 2 ) extends the geometry to a cube. Compared to the 2Dmodel, the front and back sides are additional boundaries that may influence the fluid

behavior.

Cube side length (10 cm)

Pressure point

Water

constraint (0 Pa)

Cold temperature (10 C)

on the left plate

Hot temperature (20 C)on the right plate

Figure 2: Domain geometry and boundary conditions for the 3D model (cubic cavity).

Both models calculate and compare the velocity field and the temperature field. Thepredefined Non-Isothermal Flow interface available in the Heat Transfer Moduleprovides appropriate tools to fully couple the heat transfer and the fluid dynamics.

Model Definition

2 D M O D E L

Figure 1 illustrates the 2D model geometry. The fluid fills a square cavity withimpermeable walls, so the fluid flows freely within the cavity but cannot leak out. Theright and left edges of the cavity are maintained at high and low temperatures,respectively. The upper and lower boundaries are insulated. The temperaturedifferential produces the density variation that drives the buoyant flow.

The compressible Navier-Stokes equation contains a buoyancy term on the right-handside to account for the lifting force due to thermal expansion that causes the density

variations:

u ( )u p + u u( )T +( )=23--- u( )( ) g +


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In this expression, the dependent variables for flow are the fluid velocity vector, u , andthe pressure, p. The constant g denotes the gravitational acceleration, gives the

temperature-dependent density, and is the temperature-dependent dynamic viscosity.

Because the model only contains information about the pressure gradient, it estimatesthe pressure field up to a constant. To define this constant, you arbitrarily fix thepressure at a point. No-slip boundary conditions apply on all boundaries. The no-slipcondition results in zero velocity at the wall but does not set any constraint on p.

At steady-state, the heat balance for a fluid reduces to the following equation:

Here T represents the temperature, k denotes the thermal conductivity, and C p is thespecific heat capacity of the fluid.

The boundary conditions for the heat transfer interface are the fixed high and lowtemperatures on the vertical walls with insulation conditions elsewhere, as shown inFigure 1 .

3 D M O D E L

Figure 2 shows the geometry and boundary conditions of the 3D model. The cavity isnow a cube with high and low temperatures applied respectively at the right and leftsurfaces. The remaining boundaries (top, bottom, front and back) are thermallyinsulated.

M O D E L A N A LY S I S

Before starting the simulations, it is recommended to estimate the flow regime. To thisend, four indicators are presented: the Reynolds, Grashof, Rayleigh, and Prandtlnumbers. They are calculated using the thermophysical properties of water listed inTable 1 . The thermophysical properties are given at 290 K which is in the range of thetemperatures observed in the model.

PARAMETER DESCRIPTION VALUE

Density 999 kg/m 3

Dynamic viscosity 1.0810 -3 Ns/m 2

k Thermal conductivity 0.60 W/(mK)

C p Heat capacity 4.18 kJ/(kgK)

Coefficient of thermal expansion 0.1710 -3 K-1

TABLE 1: THERMOPHYSICAL PROPERTIES FOR WATER AT 290 K

C pu T k T ( ) 0=


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Prantdl Number

Definition The Prandtl number is the ratio of fluid viscosity to thermal diffusivity. It isdefined by:

For air at 300 K, Pr = 0.71 (see, for example, A.4 in Ref. 1 ) and for water at 300 K,Pr = 5.8 (see, for example, A.6 in Ref. 1 ).

Boundary Layers The Prandtl number also indicates the relative thickness of the outerboundary layer, , and the thermal boundary layer, T. In the present case, it isreasonable to estimate the ratio /T by the relation (7.32 in Ref. 2 )

(1)

The outer boundary layer is the distance from the wall to the region where the fluidstabilizes. It is different from the momentum boundary layer, M , which measures thedistance from the wall to the velocity peak.

Application in This Model With the values given in Table 1 , the Prandtl number for water at 290 K, is found to be of the order 1 or 10. According to Equation 1 , andT should then be of same order of magnitude.

Reynolds Number

Definition The Reynolds number estimates the ratio of inertial forces to viscous forces.It is defined by the formula

where U denotes the typical velocity and L the typical length.

At atmospheric pressure and at 300 K, air and water have the following properties (A.4and A.6 in Ref. 1 ).

PARAMETER AIR WATER

1.16 kg/m3 997 kg/m3

1.8510 -5 N.s.m -2 8.5510-4 N.s.m -2

TABLE 2: THERMOPHYSICAL PROPERTIES OF AIR AND WATER AT 300 K AND ATMOSPHERIC PRESSURE

PrC p

k-----------=

T------ Pr=

Re UL

------------=


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You can thus approximate the Reynolds number by:

In these relations, U has to be in meters per second and L in meters.

The Reynolds number can be rewritten as the velocity ratio

which compares U to /( L). The latter quantity is homogeneous to a velocity and canbe seen as the typical velocity due to viscous forces.

Flow Regime The value of the Reynolds number is used to predict the flow regime.Generally, low values of Re correspond to laminar flow and high values to turbulentflow, with a critical value for the transition regime that depends on the geometry.

As an indication, Reynolds experiments concerning the flow along a straight and

smooth pipe showed that the transition regime in this case occurs when Re is between2000 and 10 4 (see chapter 1.3 in Ref. 3 ).

Momentum Boundary Layer The momentum boundary layer thickness can beevaluated, using the Reynolds number, by (5.36 in Ref. 2 )

(2)

Application in This Model The typical length L of the model is equal to 10 cm so theReynolds number is evaluated as

where U is still unknown. Estimates of this typical velocity will be provided later.

Grashof Number

Definition The Grashof number gives the ratio of buoyant to viscous forces. It isdefined by

Re air 6 104

UL Re water 106

UL

Re U L( ) -------------------=

M LRe

-----------

Re 10 5 U

Gr 2 g

2

-------------TL3

=


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where g is the gravity acceleration (equal to 9.81 m.s -2) and T is the typicaltemperature difference.

At atmospheric pressure and at 300 K, air and water have the following properties (A.4and A.6 in Ref. 1 ).

PARAMETER AIR WATER

1.16 kg/m3 997 kg/m3

1.8510 -5 N.s.m -2 8.5510-4 N.s.m -2

3.3410 -3 K-1 2.7610-4 K-1

The value of for air was here obtained by the following formula, considering the idealgas approximation:

You can then approximate the Grashof number by:

In these relations, T is given in kelvin and L in meters.

The Grashof number can also be expressed as the velocity ratio

where U 0 is defined by

(3)

This quantity can be considered as the typical velocity due to buoyancy forces.

Flow Regime When buoyancy forces are large compared to viscous forces, the regimeis turbulent; otherwise it is laminar. The transition between these two regimes isindicated by the critical order of the Grashof number which is 10 9 (see Figure 7.7 inRef. 2 ).

Application in This Model In this model, T is equal to 10 K so the Grashof numberis about 10 7 which indicates that a laminar regime is expected.

TABLE 3: THERMOPHYSICAL PROPERTIES OF AIR AND WATER AT 300 K AND ATMOSPHERIC PRESSURE

1

T ----=

Gr air 108TL

3 Gr water 4 10

9TL

3

GrU 0

2

L( ) ( )2---------------------------=

U 0 g TL=


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Table 4 provides the values of the quantities necessary to calculate U 0 . This velocity ishere of order 10 mm/s.

PARAMETER DESCRIPTION VALUE

g Gravitational acceleration 9.81 m/s 2

Coefficient of thermal expansion 0.1710 -3 K-1

Rayleigh Number

Definition The Rayleigh number is another indicator of the regime. It is defined by

so it is similar to the Grashof number except that it accounts for the thermal diffusivity,k, given by

Note: The Rayleigh number can be expressed in terms of the Prandtl and the Grashofnumbers through the relation Ra = Pr Gr .

At atmospheric pressure and at 300 K, you can use the approximations of Ra belowfor air and water (A.4 in Ref. 1 )

In these relations, T is given in kelvin and L in meters.

Using Equation 3 , the Rayleigh number can be rewritten as the velocity ratio

where the ratio / L can be seen as a typical velocity due to thermal diffusion.

Flow Regime Like the Grashof number, a critical Rayleigh value indicates thetransition between laminar and turbulent flow. For vertical plates, this limit is about

TABLE 4: THERMOPHYSICAL PROPERTIES OF WATER AT 290 K USED IN THE GRASHOF NUMBER

Ra

2 gC pk

--------------------TL3

=

k

C p----------=

Ra air 108TL

3 Ra water 2 10

10TL

3

RaU 0

2

L ( ) L ( )--------------------------------=


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109 (9.23 in Ref. 1 ).

Typical Velocity Because the viscous forces limit the effects of buoyancy, U 0 may givean overestimated typical velocity. Another approach (see 7.25 in Ref. 2 ) is to use U 1 instead, defined by

that is

(4)

or

This should be a more accurate estimate of U because the fluids thermal diffusivityand viscosity are used in the calculations. From now on, U 1 will therefore be thepreferred estimate of U .

Thermal Boundary Layer The Rayleigh number can be used to estimate the thermalboundary layer thickness, T. When Pr is of order 1 or greater, it is approximated bythe formula (7.25c in Ref. 2 )

(5)

Application in This Model Here, Ra is of order 10 8. The laminar regime is confirmedbut the Rayleigh number found is near the transition zone. The thermal boundarylayer thickness is then found to be of order 1 mm and U 1 of order 10 mm/s.

Synthesis

To prepare the simulation, it is very useful to follow the steps below that giveindications of what results to expect. It is important to remember that the quantitiescomputed here are only order of magnitude estimates, which should not be considered

with more than one significant digit.

First evaluate the Grashof and Rayleigh numbers. If they are significantly below thecritical order of 10 9, the regime is laminar. In this case, Equation 3 or Equation 4

U 1

L---- Ra=

U 1 k C p L--------------- Ra=

U 1U 0

Pr----------=

T LRa4

------------


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provide estimates of the typical velocity U that you can use to validate the model afterperforming the simulation.

According to Equation 1 , the Prandtl number determines the relative thickness of thethermal boundary layer and the outer layer. Equation 2 and Equation 5 then provideorders of magnitude of the thicknesses. When defining the mesh, refinements have tobe done at the boundary layers by, for instance, inserting three to five elements acrossthe estimated thicknesses.

Here, Gr and Ra are 10 7 and 10 8, respectively, and thus below the critical value of 10 9 for vertical plates. A laminar regime is therefore expected but because these values arenot significantly below 10 9, convergence is not straightforward. In this regime, theestimates U 0 and U 1 of the typical velocity are both of the order 10 mm/s.

For water at 290 K, Pr is about 10 so and T are of same orders of magnitude. Here,T is of the order 1 mm.

The Reynolds number calculated with U 1 is about 10 3, which confirms that the modelis close to the transition regime. Using U 1 and Equation 2 , the momentum boundarylayer thickness M is found to be about 1 mm.


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Results and Discussion

2 D M O D E L

Figure 3 shows the velocity distribution in the square cavity.

Figure 3: Velocity magnitude for the 2D model.

The regions with faster velocities are located at the lateral boundaries. The maximum velocity is 4.20 mm/s which is in agreement with the estimated typical velocity U 1 ofthe order 10 mm/s. According to Figure 4 , the momentum boundary layer thickness


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is of order 1 mm, as calculated before.

Figure 4: Velocity profile at the left boundary.


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Figure 5 shows the temperature field (surface) and velocity field (arrows) of the 2Dmodel.

Figure 5: Temperature field (surface plot) and velocity (arrows) for the 2D model.

A large convective cell occupies the whole square. The fluid flow follows theboundaries. As seen in Figure 3 , it is faster at the vertical plates where the highest

variations of temperature are located. The thermal boundary layer is of the order 1 mmaccording to Figure 6 , which is in agreement with the estimate provided by


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Equation 5 . The outer layer is slightly thicker than the boundary layer.

Figure 6: Temperature profile at the left boundary.


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Figure 7 shows the correlation between density and temperature.

Figure 7: Water density for the 2D model.

As expected, has the same profile as the temperature field. High variations are locatedat the boundaries, especially at lateral walls, and are responsible for the free convection.The variations are smoother at the center.


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3 D M O D E L

Figure 8 illustrates the velocity plot parallel to the heated plates.

Figure 8: Velocity magnitude field for the 3D model, slices parallel to the heated plates.


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A second velocity magnitude field is shown in Figure 9 . The plot is close to what wasobtained in 2D in Figure 3 .

Figure 9: Velocity magnitude field for the 3D model, slices perpendicular to the heated plates.


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In Figure 10 , velocity arrows are plotted on temperature surface at the middle verticalplane parallel to the plates.

Figure 10: Temperature (surface plot) and velocity (arrows) fields in the cubic cavity, fora temperature difference of 10 K between the vertical plates.

New small convective cells appear on the vertical planes perpendicular to the plates at

the four corners. They are more visible at lower Gr values, that is, far from thetransition regime. In Figure 11 , the temperature difference between the vertical platesis reduced to 1 K and 0.1 K to decrease the Grashof number to 10 5 and 10 6.


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Observe the bigger cells at the four corners of the plane.

Figure 11: Temperature (surface plot) and velocity (arrows) fields in the cubic cavity,with, for a temperature difference of 1 K (top) and 0.1 K (bottom) between the vertical

plates.


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Notes About the COMSOL Implementation

The material properties for water are available in the Material Library. The density anddynamic viscosity are functions of the temperature.

At high Gr values, using a good initial condition becomes important in order toachieve convergence. Moreover, a well-tuned mesh is needed to capture the solution,especially the temperature and velocity changes near the walls. Use the Stationarystudy steps continuation option with T as the continuation parameter to get a solversequence that uses previous solutions to estimate the initial condition. For this model,

it is appropriate to ramp up T

from 103

K to 10 K, which corresponds to a Grashofnumber range of 10 3107. At Gr = 103, the model is easy to solve. The regime isdominated by conduction and viscous effects. At Gr = 107, the model becomes moredifficult to solve. The regime is greatly influenced by convection and buoyancy.

To get a well-tuned mesh when Gr reaches 10 7, the element size near the prescribedtemperature boundaries has to be smaller than the momentum and thermal boundarylayer thicknesses, which are of order 1 mm according to Equation 2 and Equation 5 .

It is recommended to have three to five elements across the layers when usingP1 elements (the default setting for fluid flows).

References

1. F.P. Incropera, D.P. DeWitt, T.L. Bergman, and A.S. Lavine, Fundamentals ofHeat and Mass Transfer , 6th ed., John Wiley & Sons, 2006.

2. A. Bejan, Heat Transfer , John Wiley & Sons, 1985.3. P. A. Davidson, Turbulence: An Introduction for Scientists and Engineers , OxfordUniversity Press, 2004.

Model Library path: Heat_Transfer_Module/Tutorial_Models,_Forced_and_Natural_Convection/buoyancy_water

Modeling Instructions

From the File menu, choose New.


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N E W

1 In the New window, click the Model Wizard button.

M O D E L W I Z A R D

1 In the Model Wizard window, click the 2D button.

2 In the Select physics tree, select Fluid Flow>Non-Isothermal Flow>Laminar Flow (nitf) .

3 Click the Add button.

4 Click the Study button.

5 In the tree, select Preset Studies>Stationary .

6 Click the Done button.

G L O B A L D E F I N I T I O N S

Parameters1 On the Home toolbar, click Parameters .

2 In the Parameters settings window, locate the Parameters section.


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3 In the table, enter the following settings:

The Grashof and Rayleigh numbers should be less than 109

, indicating that alaminar regime is expected.

G E O M E T R Y 1

Square 11 In the Model Builder window, under Component 1 right-click Geometry 1 and choose

Square .

Name Expression Value Description

L 10[cm] 0.1000 m Square side length

DeltaT 10[K] 10.00 K Temperature difference

Tc 283.15[K] 283.2 K Low temperature

Th Tc+DeltaT 293.2 K High temperature

rho 999[kg/m^3] 999.0 kg/m Density

mu 1.08e-3[N*s/

m^2]

0.001080 Ns/

(mm)

Dynamic viscosity

k 0.60[W/(m*K)] 0.6000 W/(mK) Thermal conductivity

Cp 4.18[kJ/(kg*K)]

4180 J/(kgK) Heat capacity

beta 0.17e-3[1/K] 1.700E-4 1/K Coefficient of thermalexpansion

U0 sqrt(g_const*beta*DeltaT*L)

0.04083 m/s Typical velocity due tobuoyancy

U1 U0/sqrt(Pr) 0.01489 m/s Typical velocityestimation

Pr mu*Cp/k 7.524 Prandtl number

Gr (U0*rho*L/mu)^2

1.426E7 Grashof number

Ra Pr*Gr 1.073E8 Rayleigh number

Re0 rho*U0*L/mu 3777 Reynolds numberapproximation with U0

Re1 rho*U1*L/mu 1377 Reynolds numberapproximation with U1

eps_t L/(Ra)^0.25 9.825E-4 m Thermal boundary layerthickness

eps_m L/sqrt(Re1) 0.002695 m Momentum boundary layerthickness


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2 In the Square settings window, locate the Size section.

3 In the Side length edit field, type L.

4 Click the Build All Objects button.

M AT E R I A L S

On the Home toolbar, click Add Material .

A D D M AT E R I A L

1 Go to the Add Material window.

2 In the tree, select Built-In>Water, liquid .3 In the Add material window, click Add to Component .

N O N - I S O T H E R M A L F L O W

Define the initial temperature as the mean value between the high and low temperature values.

Initial Values 11 In the Model Builder window, under Component 1>Non-Isothermal Flow click Initial

Values 1 .

2 In the Initial Values settings window, locate the Initial Values section.

3 In the T edit field, type (Tc+Th)/2 .

4 In the p edit field, type rho*g_const*(L-y) .

This setting gives an initial pressure field consistent with the volume force. The

initial pressure field must be also consistent with the pressure constraint (see thesection Pressure Point Constraint below).

Volume Force 11 On the Physics toolbar, click Domains and choose Volume Force .

2 Select Domain 1 only.

3 In the Volume Force settings window, locate the Volume Force section.

4 Specify the F vector as

In this expression for the buoyancy flow, g_const is a predefined constantcorresponding to the gravity acceleration.

0 x

-nitf.rho*g_const y

l d h l h


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Pressure Point Constraint 11 On the Physics toolbar, click Points and choose Pressure Point Constraint .

2 Select Point 2 only.Fixing the pressure at an arbitrary point is necessary to define a well-posed model.

Temperature 11 On the Physics toolbar, click Boundaries and choose Temperature .

2 Select Boundary 1 only.

3 In the Temperature settings window, locate the Temperature section.

4 In the T 0 edit field, type Tc .




4 In the T 0 edit field, type Th .

Now modify the default mesh size settings to ensure that the mesh satisfies thecriterion discussed in the Introduction section.

M E S H 1

1 In the Model Builder window, under Component 1 click Mesh 1 .

2 In the Mesh settings window, locate the Mesh Settings section.

3 From the Element size list, choose Extra fine .

4 Click the Build All button.

S T U D Y 1

Because the Grashof number is near the critical value of around 10 9, the model ishighly nonlinear. To achieve convergence, use continuation to ramp up thetemperature difference value from 10 -3 K to 10 K, which corresponds to a Grashofnumber from 10 3 to 10 7.

Step 1: Stationary 1 In the Model Builder window, under Study 1 click Step 1: Stationary .

2 In the Stationary settings window, click to expand the Study extensions section.

3 Locate the Study Extensions section. Select the Auxiliary sweep check box.

4 Click Add.

Solved with COMSOL Multiphysics 4 4


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6 In the Model Builder windows toolbar, click the Show button and select AdvancedPhysics Options in the menu.

The pseudo stepping time option is generally useful to help the convergence of astationary flow model. However, a continuation approach is already used here. Inthis precise model, disabling the pseudo time stepping option improves the

convergence. Follow the instructions below to do so.

N O N - I S O T H E R M A L F L O W

1 In the Model Builder window, under Component 1 click Non-Isothermal Flow .

2 In the Non-Isothermal Flow settings window, click to expand the Advanced settings section.

3 Locate the Advanced Settings section. Find the Pseudo time stepping subsection.

Clear the Use pseudo time stepping for stationary equation form check box.

S T U D Y 1

On the Home toolbar, click Compute .

R E S U LT S

Velocity (nitf)

The first default plot group shows the velocity magnitude as in Figure 3 . Notice thehigh velocities near the lateral walls due to buoyancy effects.

Temperature (nitf)The second default plot shows the temperature distribution. Add arrows of the velocityfield to see the correlations between velocity and temperature, as in Figure 5 .

1 On the 2D plot group toolbar, click Plot .

2 In the Model Builder window, under Results right-click Temperature (nitf) and chooseArrow Surface .

3 In the Arrow Surface settings window, locate the Coloring and Style section.

4 From the Color list, choose Black .


To reproduce the plot of Figure 7 , follow the steps below.

Auxiliary parameter Parameter value list

DeltaT 1e-3 1e-2 1e-1 1 10

Solved with COMSOL Multiphysics 4 4


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2D Plot Group 31 On the Home toolbar, click Add Plot Group and choose 2D Plot Group .

2 In the Model Builder window, under Results right-click 2D Plot Group 3 and chooseSurface .

3 In the Surface settings window, locate the Expression section.

4 Click Density (nitf.rho) in the upper-right corner of the section. In the Model Builder window, right-click 2D Plot Group 3 and choose Rename .

5 Go to the Rename 2D Plot Group dialog box and type Density in the New name editfield.

6 Click OK.


In the following steps, the temperature and velocity profiles are plotted near the leftboundary in order to estimate the boundary layer thicknesses of the solution.

Data Sets1 On the Results toolbar, click Cut Line 2D .

2 In the Cut Line 2D settings window, locate the Line Data section.

3 In row Point 1 , set x to 0 .

4 In row Point 1 , set y to 5[cm] .

5 In row Point 2 , set x to 1[cm] .

6 In row Point 2 , set y to 5[cm] .

7 Click the Plot button.


2 In the 1D Plot Group settings window, locate the Data section.

3 From the Data set list, choose Cut Line 2D 1 .

4 From the Parameter selection (DeltaT) list, choose Last .

5 On the 1D plot group toolbar, click Line Graph .


7 In the Model Builder window, right-click 1D Plot Group 4 and choose Rename .

8 Go to the Rename 1D Plot Group dialog box and type Temperature at boundarylayer in the New name edit field.



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p y


9 Click OK.

The thermal boundary layer is around 3 mm.



3 From the Data set list, choose Cut Line 2D 1 .

4 From the Parameter selection (DeltaT) list, choose Last .

5 On the 1D plot group toolbar, click Line Graph .

6 In the Line Graph settings window, locate the y-axis data section.

7 Click Velocity magnitude (nitf.U) in the upper-right corner of the section. On the 1Dplot group toolbar, click Plot .


9 Go to the Rename 1D Plot Group dialog box and type Velocity at boundarylayer in the New name edit field.

10 Click OK.The momentum boundary layer is around 1 mm and the outer layer between 5 mmand 10 mm.

Now create the 3D version of the model.

R O O T

On the Home toolbar, click Add Component and choose 3D.

G E O M E T R Y 2

On the Home toolbar, click Add Physics .

A D D P H Y S I C S

1 Go to the Add Physics window.

2 In the Add physics tree, select Recently Used>Non-Isothermal Flow (nitf) .

3 In the Add physics window, click Add to Component .

R O O T

On the Home toolbar, click Add Study .

A D D S T U D Y

1 Go to the Add Study window.



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p y


2 Find the Studies subsection. In the tree, select Preset Studies>Stationary .

3 In the Add study window, click Add Study .

G E O M E T R Y 2

Block 11 On the Geometry toolbar, click Block.

2 In the Block settings window, locate the Size section.

3 In the Width edit field, type L.

4 In the Depth edit field, type L/2 .5 In the Height edit field, type L.

6 Click the Build All Objects button.

A D D M AT E R I A L

1 Go to the Add Material window.

2 In the tree, select Built-In>Water, liquid .

3 In the Add material window, click Add to Component .

N O N - I S O T H E R M A L F L O W 2

Initial Values 11 In the Model Builder window, under Component 2>Non-Isothermal Flow 2 click Initial

Values 1 .

2 In the Initial Values settings window, locate the Initial Values section.

3 In the T 2 edit field, type (Tc+Th)/2 .

4 In the p2 edit field, type rho*g_const*(L-z) .

Volume Force 11 On the Physics toolbar, click Domains and choose Volume Force .

2 Select Domain 1 only.

3 In the Volume Force settings window, locate the Volume Force section.4 Specify the F vector as

0 x

0 y

-nitf2.rho*g_const z



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Pressure Point Constraint 11 On the Physics toolbar, click Points and choose Pressure Point Constraint .

2 Select Point 4 only.Temperature 11 On the Physics toolbar, click Boundaries and choose Temperature .



4 In the T 0 edit field, type Tc .




4 In the T 0 edit field, type Th .

Symmetry, Heat 11 On the Physics toolbar, click Boundaries and choose Symmetry, Heat .


Symmetry, Flow 21 On the Physics toolbar, click Boundaries and choose Symmetry, Flow .


M E S H 2To obtain reliable results in a reasonable computational time, create a structured meshaccording to the steps below.

Mapped 11 In the Model Builder window, under Component 2 right-click Mesh 2 and choose

Mapped .


Distribution 11 Right-click Component 2>Mesh 2>Mapped 1 and choose Distribution .

2 Select Edges 1, 3, 5, and 9 only.

3 In the Distribution settings window, locate the Distribution section.

4 From the Distribution properties list, choose Predefined distribution type .



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5 In the Number of elements edit field, type 16 .

6 In the Element ratio edit field, type 3 .

7 Select the Symmetric distribution check box.8 Click the Build All button.

The front face mesh has smaller elements near the edges because large variations in velocity

and temperature are expected there.Now extend the front mesh to the remaining structure.

Swept 1In the Model Builder window, right-click Mesh 2 and choose Swept .

Distribution 11 In the Model Builder window, under Component 2>Mesh 2 right-click Swept 1 and

choose Distribution .2 In the Distribution settings window, locate the Distribution section.

3 From the Distribution properties list, choose Predefined distribution type .

4 In the Number of elements edit field, type 8 .

5 In the Element ratio edit field, type 3 .



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6 Select the Reverse direction check box.

To resolve the boundary layers, use a Boundary Layers feature to generate smaller mesh

elements near the walls.Boundary Layer Properties1 In the Model Builder window, right-click Mesh 2 and choose Boundary Layers .

2 Select Boundaries 1 and 36 only.

3 In the Boundary Layer Properties settings window, locate the Boundary LayerProperties section.

4 In the Number of boundary layers edit field, type 5 .5 From the Thickness of first layer list, choose Manual .

6 In the Thickness edit field, type eps_t/5 .

7 Click the Build Selected button.

S T U D Y 2

Step 1: Stationary 1 In the Model Builder window, under Study 2 click Step 1: Stationary .

2 In the Stationary settings window, click to expand the Study extensions section.

3 Locate the Study Extensions section. Select the Auxiliary sweep check box.

4 Click Add.


6 Locate the Physics and Variables Selection section. In the table, enter the followingsettings:

N O N - I S O T H E R M A L F L O W 2

1 In the Model Builder window, under Component 2 click Non-Isothermal Flow 2 .

2 In the Non-Isothermal Flow settings window, click to expand the Advanced settings section.

Auxiliary parameter Parameter value listDeltaT 1e-3 1e-2 1e-1 1 10

Physics Solve for Discretization

Non-Isothermal Flow physics



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3 Locate the Advanced Settings section. Find the Pseudo time stepping subsection.Clear the Use pseudo time stepping for stationary equation form check box.

S T U D Y 2

Solver 21 On the Study toolbar, click Show Default Solver .

2 In the Model Builder window, expand the Study 2>Solver Configurations>Solver2>Stationary Solver 1 node.

3 Right-click Direct and choose Enable .

4 In the Direct settings window, locate the General section.

5 From the Solver list, choose PARDISO.

6 On the Home toolbar, click Compute .

R E S U LT S

Velocity (nitf2)

This default plot group shows the fluid velocity magnitude in only half of the cube. Toplot the other half, proceed as follows.

Data Sets1 On the Results toolbar, click More Data Sets and choose Mirror 3D .

2 In the Mirror 3D settings window, locate the Plane Data section.

3 From the Plane list, choose zx-planes .

A new data set containing mirror values is now created. Return to the velocity plot touse this data set.

Velocity (nitf2)1 In the Model Builder window, under Results click Velocity (nitf2) .


3 From the Data set list, choose Mirror 3D 1 .

4 On the 3D plot group toolbar, click Plot .Temperature (nitf2)This default plot group shows the temperature distribution. The mirror data setcreated previously can be reused here to plot the entire cube.




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4 Right-click Results>3D Plot Group 8 and choose Slice.5 In the Slice settings window, locate the Expression section.

6 Click Velocity magnitude (nitf2.U) in the upper-right corner of the section. Locate thePlane Data section. From the Plane list, choose zx-planes .


8 Go to the Rename 3D Plot Group dialog box and type Velocity, front plane inthe New name edit field.

9 Click OK.


This slice view shows the velocity magnitude in the same plane as in the 2D model(Figure 9 ).

Next, plot arrows of the tangential velocity field in the vertical plane parallel to theplates to reproduce Figure 10 .




4 Right-click Results>3D Plot Group 9 and choose Slice.

5 In the Slice settings window, locate the Plane Data section.

6 In the Planes edit field, type 1 .

7 Locate the Coloring and Style section. From the Color table list, choose ThermalLight .


9 In the Model Builder window, right-click 3D Plot Group 9 and choose Arrow Volume .

10 In the Arrow Volume settings window, locate the Expression section.

11 In the x component edit field, type 0 .

12 In the y component edit field, type v2 .

13 In the z component edit field, type w2.

14 Locate the Arrow Positioning section. In the Points edit field, type 1 .

15 In the Points edit field, type 25 .

16 In the Points edit field, type 25 .


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